Theorem sprel 48659

Description: An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)

Assertion

Ref	Expression
sprel	⊢ (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏})

Distinct variable groups:   𝑉,𝑎,𝑏   𝑋,𝑎,𝑏

Proof of Theorem sprel

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6920	. 2 ⊢ (𝑋 ∈ (Pairs‘𝑉) → 𝑉 ∈ V)
2		sprvalpw 48655	. . . 4 ⊢ (𝑉 ∈ V → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
3	2	eleq2d 2846	. . 3 ⊢ (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) ↔ 𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}))
4		eqeq1 2764	. . . . . 6 ⊢ (𝑝 = 𝑋 → (𝑝 = {𝑎, 𝑏} ↔ 𝑋 = {𝑎, 𝑏}))
5	4	2rexbidv 3227	. . . . 5 ⊢ (𝑝 = 𝑋 → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}))
6	5	elrab 3644	. . . 4 ⊢ (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ↔ (𝑋 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}))
7	6	simprbi 503	. . 3 ⊢ (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏})
8	3, 7	biimtrdi 256	. 2 ⊢ (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}))
9	1, 8	mpcom 39	1 ⊢ (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏})

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  {crab 3412  Vcvv 3450  𝒫 cpw 4556  {cpr 4586  ‘cfv 6538  Pairscspr 48652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-spr 48653

This theorem is used by:  prssspr  48660  prsprel  48662  reupr  48697